METHOD AND SYSTEM FOR OPTICAL MEASUREMENTS FIELD OF THE INVENTION

This invention relates to method and system for the Polarized Spectral Imaging (further below PSI) for use in measuring/inspection systems. BACKGROUND OF THE INVENTION

Various types of optical systems for spectral analysis of light use two basic physical approaches either the phenomenon of angular separation or interference of light.

Phenomenon of light interference particularly used for optical imaging based on Fourier transform spectroscopy (FTS). Fig. 1 illustrates basic type of Fourier filter (further FF) comprises two polarizers 10 and 11 and active element 20 (e.g. phase delay element) placed between polarizer 10 and 20. Polarizers 10 and 11 could be oriented crossed or parallel to each other. As active element 20 could be adjustable birefringence crystal with optical axis, oriented at 45 degrees relative to polarizers and producing adjustable optical path difference (OPD) between ordinary and extraordinary beams. Spectral transmittance of FF Filter could be expressed as: r ^pF (ω) = (l + cos*»τ) - parallel oriented polarizers. (2.1.1)

= (1 - cos ωτ) - crossed oriented polarizers. (2.1. 2) τ(ω) - difference between active element passing time for ordinary and extraordinary beams. Interferograms for parallel and crossed oriented polarizers for all schemes of 2.1.1., except double pass scheme (see 2.1.1.5), can be expressed as:

+ ^cos a>τ)dω - parallel oriented polarizers. (2.1.3)

/ ₊ (? ^■ ) = (ω)(l - cosωτ)dω - crossed oriented polarizers. (2.1.4)

S _Sysl (ω) - measurement system spectrum

R- _Target ( ^ω ) " measured sample reflectance spectrum

SUMMARY OF THE INVENTION

According to the invention Polarized Spectral Imaging is carried out in optical schemes with three basic configurations: • Spatial Imaging systems.

• Angle Imaging systems.

• Combined Imaging systems.

Spatial Imaging systems allow to receive information on sample (wafer) reflectance, and therefore about sample structure, for each of the point in the field of view (spatial image). Angle Imaging systems allow receiving information about sample (wafer) reflectance or diffraction, for each of the angle in the angular field of view (angular image), averaged on measured spot.

Combined Imaging systems allow receiving the both types of information.

In each of the three general configurations PSI can implement measurement of the following sample reflectance / diffraction characteristics:

• Sample reflectance interferograms and reflectance spectrum, that couls be extracted from reflectance interferograms by special algorithm, for both polarization directions.

Interferogram is the direct signal of measurement system, and can be expressed as:

/(T) = ]s _Sysl (ω)R _Taigeι {ω\l + cos ωτ)dω (1.1) o

S _Syst (ώ) - measurement system spectrum

R _Targel ( ^ω ) - measured sample reflectance spectrum.

Sign "+" or "-" in equation (1.1) depends on system configuration.

• Sample reflectance Differential Interferogram (DI), that contains information about phase difference between reflectance of two different polarizations. DI can be expressed as:

/(«) = [S ₀ (ω) + S _e (ω) ± 2p _o (ω)S _e (ω) cos Aφ _oe (ω, u)]dω (1.2) o

S ₀ (ω) = Ss _yst (CO)R _T3X ^ _I ( ^ω ) ~ ligh-t intensity spectrum for ordinary beam

S _e (ώ) = S _S ^e _ysl (co)Rγ _arset (ω) - light intensity spectrum for extraordinary beam

Aφ _oe (ω, u) - phase difference between ordinary and extraordinary beams, reflected from sample, as function of angle frequency and active element scanning parameter. Sign "+" or "-" in equation (1.2) depends on system configuration.

All schemes, described bellow, implement measurement one ore several reflectance characteristics for various general configurations.

DESCRIPTION OF PARTICULAR EMBODIMENTS

Spatial Imaging Measurement Schemes.

Fig. 2 shows an optical shemes for Reflectance Spectrum Measurement based on FF Filter 200 located close to Imaging Sensor 60. Light from a light source 30 is collimated by a relay lens 42 and further reflected by a beam splitter 50 and focused on the sample 100 (e.g. wafer with patterned structures forming diffracting grating) by an objective lens 40. The diffracted light is than re-collimated by objective lens 40, passed through FF Filter 20 and finally focused on an imaging sensor 60 (e.g. CCD or MosFET two-dimensional sensor array) by tube lens 41.

Fig. 2A illustrates another embodiment of optical system similar to system of Fig. 2 but using reflective optics objective. In the present embodiment light from light source 30 passes through Beam Splitter 50 than reflected from the mirror 70 and focused on the sample by a concave mirror 71. Returned from sample 100 light is further reflected by concave mirror 71 and mirror 70 towards beam splitter 50 then passes through the FF Filter 200 and finally focused on the imaging sensor 60.

According to another embodiment of the present invention as further show on Fig. 3, FF Filter 200 could be positioned between light source 30 and sample 30. In that case light is collimated by a lens 42 and passes through FF Filter 200 then it is reflected by a beam splitter 50 and focused on sample (wafer) 100 by an objective lens 40. Returned from the sample 100 light is re-collimated by the objective 40 and finally focused on an imaging sensor 60 by tube lens 41.

_ , which could be connected to a Measurement Unit (MU) by optical fiber 80. Light is collimated by a lens 42 and passes through FF Filter 200 then focused by an optical fiber entrance relay 43. An optical fiber 80 transfers the Fourier modulated light from IU to MU and focused on sample (wafer) 100 by an objective lens 40. Returned from the sample 100 light is re-collimated by the objective 40 and finally focused on an imaging sensor 60 by tube lens. Various configurations of MU could be implemented and Fig. 3 A exemplifies one of them. Optical fiber exit relay lens 44 collimates light received from IU by fiber 80. Light is reflected from beam splitter 50 and focused on the sample (wafer) 100 by the objective lens 40. The diffracted light (returned from the sample 100 is then re-collimated by the objective lens 40 passes trough beam splitter 50 and finally focused on the imaging sensor 60.

As further illustrated on Fig. 4 FF Filter 200 could be positioned between sample 100 and imaging sensor 60 in such a way that light passes FF filter in two way. Light is first collimated by lens 42, then reflected from a beam splitter 50, and further passed through FF Filter 200 (first pass), and focused on the sample 100 by an objective lens 40. The returned (diffracted) light is then re- collimated by the objective lens 40 and passed through the FF Filter 200 (second pass) and finally focused on the imaging sensor 60 by tube lens 41, thereby light undergoes

Fourier filtration twice.

Double pass Fourier filter transmittance for parallel and crossed oriented polarizers (non polarized light) can be expressed as:

7J (ω) = — (1 + cos ωτ) - parallel oriented polarizers. (2.1.5) 8

T° ^p (ω) = — (1 - cos ωτ) ² - crossed oriented polarizers. (2.1.6) 8 τ{ω) - difference between active element passing time for ordinary and extraordinary beams. Interferograms for double pass with parallel and crossed oriented polarizers can be expressed as:

ψ (τ) - parallel oriented polarizers. (2.1.7)

1 °° jv ^p ( _r ) = _ js _Sysi (ω)R _Taxse , (α>)(l - cos ωτ) ² dω - crossed oriented polarizers. (2.1.8)

^• _Target ( ^ω ) ' measured sample reflectance spectrum

Using both interferograms, measured with parallel and crossed polarizers orientation gives the following results:

/,r (r) + /f (r) = i If (2τ) + If (O) (2.1.9)

/,r(r)-/rω = 2(/-(r)-/-(0)) (2.1.10)

The (2.1.9) expression represents interferogram with double scanning range of active element - if (2τ) . It allows to receive doubled delay, produced by the same active element.

The (2.1.10) expression represents the regular interferogram of single pass - lf (τ)

According to still another embodiment, FF filter could include only single, polarizer. In that case, as further illustrated on Fig. 5, type of FF Filter comprises a single polarizer P, abeam splitter BS, a mirror MR and active element AE. Active element AE is oriented at 45 degrees to polarizer P and produces adjustable optical path difference (OPD) between ordinary and extraordinary beams. Incoming light is first passed through beam splitter BS, polarizer P and active element AE, then reflected by mirror MR, then passed second time through active element AE and polarizer P, and further reflected from beam splitter BS towards detector (not shown.

It is possible only one type of spectral transmittance, equivalent to parallel oriented polarizers (see two polarizes scheme - 2.1.1.):

T ^F (ώ) = — (1 + cosωτ) - parallel oriented polarizers. (2.1.11)

τ(ω) - difference between active element passing time for ordinary and extraordinary beams.

Interferogram is equivalent to parallel oriented polarizers (see two polarizes scheme - 2.1.1.):

I(r) = ^)s _Sysl (ω)R _Tareel (ω)(l + cosωr)dω (2.1.12)

⁴ O S _Sysl (ω) - measurement system spectrum

- _™ , _V v ^■ measure samp e re ec ance spec rum

Fig. 6 illustrates an optical scieme based on single polarizer FF Filter 200 'located between sample 100 and imaging sensor 100. In that case light from light source 30 is first collimated by relay lens 42, then reflected from a beam splitter 50 and focused on sample (wafer) 100 by an objective lens 40. The returned (diffracted) light is then re-collimated by objective lens 40, passes through FF filter 200' comprising polarizer 10 and active element 20, focused on mirror 70 by relay lens 45, then light reflected from mirror 70 is re-collimated by lens 45, passed second time through FF filter 200'. Then light is reflected from beam splitter 51, and finally focused on an imaging sensor 60 by lens 41.

Fig. 7 further illustrates an optical scieme based on single polarizer FF Filter 200' located between light source and sample. In that case light from a light source 30 is first collimated by lens 42 and passes through polarizer 10 and active element 20 of FF filter 200'. Then it is focused on mirror 70 by lens 45, reflected from mirror 70 and re-collimated by lens 45, then passes second time through active element 20 and polarizer 10 of FF filter 200', then reflected from a first beam splitter 50 and second beam splitter 51 and focused on a sample (wafer) 100 by an objective lens 40. The returned from sample (diffracted) light is then re-collimated by objective lens 40 and finally focused on an imaging sensor 60 by tube lens 41.

Similar to embodiment of Fig. 3 A, FF Filter could be positioned inside separate illumination unit as further illustrated by Figt. 7A. In this case system comprises of two parts - Measurement Unit (MU) and separate Illumination Unit (IU) and operates similar to embodiment of Fig. 7. Optical fiber 80 transfers light from IU to MU. Light could be collimated by optical fiber entrance relay lens 43 and re-collimated by optical fiber exit relay lens 44. It is clear that MU could be implemented in various ways and Fig. 7 A exemplifies only one of them. Fig. 8 futher iluustrates another embodiment of the present invention using differential interferometric (DI) shemes. In contrast to previously described reflectance spectrum measurement schemes, differential interferometric shemes include sample mirror as part of interferometer and allows to receive a spatial image of interference result of two polarized ortogonaly light beams, reflected from optical anysotrophic sample including reflectance spectrums and phase difference between different polarised reflection. It could provide an additional measurement information. In case of isotrophic sample (solid), measurement result of

. be used directly as the special measurement signal can be used directly.

Another way is extracting phase difference between ordinary and extraordinary beams, reflected from sample, from DI and using it as additional measurement information. Basic implementation of DI measurements.

As illustrated on Fig. 9, light from a light source 30 is collimated by a lens 42, then passes through a polarizer 10, reflected from a beam splitter 50 and focused on a sample (wafer) 100 by an objective lens 40. The returned from sample 10 (diffracted) light is then re-collimated by objective lens 40, then passes through an active element 20 and polarizer 11, and finally focused by a lens 41 on an imaging sensor 60. This scheme implements both configuration of direct signal (see eq. (1.2)) "+" for parallel oriented polarizers and "-"for crossed oriented polarizers.

As further illustrated on Fig. 10 DI technique also could be implemented on optical scieme with single polarizer. In that example light is first collimated by lens 42, reflected from a beam splitter 50, then passes through a polarizer 10 and an active element 20 and focused on the sample (wafer) 10 by an objective lens 40. The returned (diffracted) light is further re-collimated by the objective lens 40, then passes second time through active element 20 and polarizer 10 and finally focused by tube lens 41 on an imaging sensor 60.

This scheme is the double pass scheme and supports only "+" configuration of direct signal (see l., eq. (1.2)).

Extracting Phase Difference.

In order to extract phase difference between ordinary and extraordinary beams, reflected from sample, from DI (term 1.2) the invented method requires to perform the following stages: a. Measure light intensity spectrum for ordinary beam S ₀ (ω) .

b. Measure light intensity spectrum for extraordinary beam S _e (ω) . c. Measure DI. d. Extract phase difference from DI.

r . rs c , en Polarizer 10, an active element 20, a polarizer 11, then reflected from a beam splitter 50 and focused on a sample (wafer) 100 by an objective lens 40. The returned (diffracted) light is then re-collimated by objective lens 40 passes through a second active element 21, a polarizer 12 and finally focused by lens 41 on an imaging sensor 60.

Light intensity spectrum measurement for ordinary and extra ordinary beams is performed as the follows:

• Active element 21 is set to zero OPD.

• Polarizer 12 is oriented parallel to active element optical axis. • Measure by Fourier Spectrometer the S ₀ (ω) (Active Element 1 scanning).

• Orient Polarizer 12 at 90° relative to active element optical axis.

• Measure by Fourier Spectrometer the S _e (ω) (active element 20 scanning).

Angular Imaging Measurement Schemes.

The mam difference between angular PSI and spatial PSI schemes is the imaging sensor location. In order to measure angular distribution of light, reflected (or diffracted) from measured sample, imaging sensor is placed at back focal plane (Fourier plane), of objective lens

(see Fig. 11). Spatial PSI schemes, unlike angular, have additional imaging lens, in order to create spatial image on imaging sensor (see Fig. 2).

Thereby, all schemes, previously described could be used also for angular imaging configuration. The following schemes illustrate implementation of angular reflectance spectrum and DI measurement at PSI in accordance with additional aspects of the invention.

Fig. 11 illustrates example optical scheme for angular reflectance spectral measurement.

Light from illuminating source 30 is collimated by a light source relay lens 42 and passes through FF Filter 200 and further reflected by a beam splitter 50 and focused on a sample (wafer) 100 by objective lens 40. The returned from sample 100 (diffracted) light creates angular image on the imaging sensor 60, placed at back focal plane of the objective lens 40.

Fig. 12 further illustrates another embodiment of the present invention, where light from illuminating source 30 is collimated by a light source relay lens 42 and passes through a first polarizer 10 and an active element 20, then it reflected by a beam splitter 50 and focused on a

. u uUIr-ICIeU Hg passed back through objective lens 40, bam splitter 50 and second polarizer 22, and then creates an angular image on an imaging sensor 60, placed at back focal plane of the objective lens 40.

Combined Spatial and Angular PSI Schemes.

Combined schemes that support both imaging modes - spatial and angular by the same measurement device could be used.

Figs. 13 A and 13B illustrate operational principles of combined systems in different imaging modes. For simplifying, they doesn't contain illuminating part (light source, beam splitter, etc.), and show only diffracted light path from sample to imaging sensor.

• Fig. 13 A illustrates the Spatial Imaging mode of combined systems. Light, returned (diffracted) from sample, is re-collimated by Objective Lens (OL) and focused by Relay Lens (RL) on Imaging Sensor. In such a way the spatial image of sample is created on the Imaging Sensor. • Fig. 13Billustrates the Angular Imaging mode of combined systems. The angular image of light, diffracted from sample, first is created on Back Focal Plane of the OL. The RL observes this image and duplicates it on the Imaging Sensor.

Combined systems could be generally implemented in two ways: a. By using two different imaging sensor - the first one for angular imaging, and the second one for spatial imaging. It allows to perform the both imaging measurements simultanesly and don't require optical reajustment for imaging mode selection. b. Schemes, contain one imaging sensor for both modes. They require optical reajustment for imaging mode selection.

Fig. 14 illustrates a combined optical scheme using two imaging sensors.

Light from a illuminating source 30 is collimated by light source relay lens 42 and passes through FF Filter 200, then reflected by a first beam splitter 50 and focused on a sample (wafer) 100 by an objective lens 40. Light returned from sample 100 (diffracted) passes through first beam splitter 50 and creates angular image on the back focal plane of the objective lens 40. Then light is divided into two parts by a second beam splitter 51. The first part of light is reflected by second beam splitter 51 and passes through a relay lens 41b to a first imaging sensor 60b. Relay lens 41 duplicates the angular image, created on the back focal plane of the objective lens, 40 to second imaging sensor 60 (angular imaging).

The second part is passed through second beam splitter 51 and diaphragm 90 (aperture stop), and finally focused by a relay lens 41 on second imaging sensor 60, placed on focus plane of the relay lens 41 (spatial imaging).

The diaphragm 80 limits the angular distribution of diffracted light for spatial imaging. In accordance with another embodiment of the present invention, combined optical scheme with single imaging sensor could be used. In that case adjustable position of relay lens provides combined spatial and angular PSI modes of operation.

As further illustrated on Fig. 15 light is collimated by light source relay lens 42 and passes through FF Filter 200 then reflected by a beam splitter 50 and focused on a sample (wafer) 100 by an objective lens 40. Returned from sample 100 (diffracted) light passes through abeam splitter 50 and creates angular image on the back focal plane of the objective lens 40, then the light is focused by movable relay lens on imaging sensor 60. By adjusting the movable relay lens MRL position along optical axis, it is possible to switch imaging mode between angular and spatial. In order to limit angular distribution of diffracted light for spatial mode, the diaphragm 90 could be removed or adjusted to smaller aperture.

In accordance with still another embodiment of the present invention, combined scheme using removable relay lens MRL and diaphragm could be implemented as illustrated on Fig. 15 A. In accordance with this embodiment light is collimated by light source relay lens 42 and passed through FF Filter 200, then reflected by a beam splitter 50 and focused on a sample (wafer) 100 by an objective lens 40. The diffracted light passes through beam splitter 50.

of optical path, and the angular image is created on imaging sensor 60, placed on the back focal plane 300 of objective lens 40.

• In spatial imaging mode relay lens 4 land diaphragm 90 are moved in, and spatial image is created by relay lens 41 on the imaging sensor 60.

Schemes for Transmittance Measurement.

Fig. 16 further illustrates embodiment providing a transmittance measurement technique in accordance with the present invention. Light is collimated by light source relay lens 42 and passes through FF Filter 200 then focused on a sample 100 by objective lens 40. The transmitted light is than re-collimated by second objective lens 40b and finally focused on an imaging sensor 60 by relay lens 41.

Fig. 17 illustrates an example of angular transmittance spectral imaging.

Light is collimated by light source relay lens 42 and passes through FF Filter 200 then focused on a sample 100 by objective lens 40. Transmitted light creates angular image on the imaging sensor 60 placed at back focal plane of the lens 40b.

According still another embodiment additional spectrometric channel could be used as illustrated on Fig. 18 A. Additional spectrometric channel could be used for performing following functions: a) Active Element Phase Delay Calibration b) Correction of active element actual OPD c) Light source instability correction

Spectrometer 400, placed before FF filter 200, provides only function a), while placed after FF filter 200 it performs all three functions.

Fig. 18 A exemplifies spectrometer positioned after FF filter 200 and Fig. 18B illustrates embodiment with spectrometer before FF filter 200.

Active Element Actual OPD Correction.

In order to change OPD during measurement cycle (scanning), active element undergoes readjustment of its scanning parameter, for example, SB compensator prism position or Pockels cell applied voltage. Additional reference spectrometer, placed after FF filter (for example see

, parameter actual value.

The FF filter transmittance, as function of wave length and scanning parameter value - u, can be expressed as: T(λ,u) = l ± cosφ(λ,u) (6.1)

"+" for parallel polarizers orientation, "-" for crossed orientation. φ(λ, u) - phase delay, produced by active element.

Therefore, spectrum, measured by reference spectrometer can be expressed as: S{λ, u) = S ₀ {λ) • (1 ± cos φ{λ, «)) , (6.2)

where S ₀ (λ) - considers light source spectrum, spectrometer response spectrum, and transmittance spectrum of all optical elements between light source and spectrometer.

Graph of Fig. 19 shows experimental spectrum, measured by spectrometer with two different scanning parameter values.

Wave lengths, ensure spectrum local minimums correspond to phase delays, should satisfy to the following condition: φ{pι _k ,U _x ) = π(2k + 1) - for parallel polarizers orientation (6.3)

φ{X _k ,U _x ) - 2πk - for crossed polarizers orientation (6.4)

In general algorithm of scanning parameter actual value correction looks as the following:

• Set scanning parameter to desirable value. • Read reference spectrometer and imaging sensor data.

• Find wave lengths /ζ , ensure local minimums of measured spectrum, closed to zero (see graph on Fig. 19).

• Find actual u _x ensures optimal fitting to active element calibration data for found X _k .

, u p _k , doesn't depend on system spectrum S ₀ {λ) , because if S ₀ (λ) ≠ 0 measured spectrum can equal zero only if T(λ,u) = 0.

Scheme with Internal Reference Mirror. In accordance with another broad aspect of the present invention, schemes with internal reference mirror could be used, for example, as shown on Fig. 20, creating on the same imaging sensor two images — sample and reference, allow perform the following advanced operation:

• Light source and imaging sensor instability correction per each imaging sensor measurement. The advantage of using the same detector is correction of light source jumps and imaging sensor read noise.

• Internal system calibration with known reference mirror reflectance spectrum, based on Spectrum Reconstruction Algorithm, that doesn't require mechanical readjustment (motion). In that case, light is collimated by light source relay lens 42 and passes through FF Filter 200 then focused on a sample 100 by objective lens 40 then reflected by a beam splitter 50 and focused on reference mirror FM, contained hole H. Reference mirror RM divides light into two parts. The first part of light is reflected from reference mirror RM, than re-collimated by lens 45, passes through beam splitter 50 and finally focused by lens 41 on imaging sensor 60. In such way the reference image is created on imaging sensor 60.

The second part of light is passed throw the hole H of reference mirror RM, then re-collimated by lens 45, and focused by objective lens 40 on sample 10. Then diffracted light is re-collimated by lens 40, focused by lens 45b on reference mirror plane 45, passes through hole H of reference mirror RM, then collimated by lens 45, passes through beam splitterSO and finally focused by lens 41 on imaging sensor 60. In such way the sample image is created on imaging sensor 60.

Fig. 21 illustrates another embodiment of the present invention using combination of reference spectrometer and imaging sensor reference zone. Light is transferred via optical fiber 80 from light source 30 and collimated by light source relay lens 44, then reflected by the beam splitter 50 and focused on sample (wafer) 10 by the objective lens 40. Returned (diffracted) light passes through beam splitter 50, and creates angular image on the back focal plane of the objective lens 40, then the light is focused by movable along optical axis lens ML on the virtual imaging plane

_</ , or angular image on the virtual imaging plane VIP. Then light is collimated by lens 45 and passed through filter FF 200. A tube lens 41 further reconstructs the image (spatial or angular), created on virtual imaging plane VIP, on a pinhole mirror 75. One part of light, passed through pinhole 76 is measured by reference spectrometer 400. The other part, reflected by mirror 75, is finally focused by relay lens 45 on measurement area MA of imaging sensor 60.

Adjustable diaphragm 90, allows limit the angular distribution of diffracted light in spatial mode, and the spot size in spatial mode. For imaging sensor and light source instability correction an additional light channel could be used. In that case light passed through beam splitter 50, could be focused on optical fiber 80b by lens 43. The second end of optical fiber 80b is placed on the virtual imaging plane VIP. Light, passed through the optical fiber 80b, is collimated by lens 43, passed through FF filter 200, focused on the pinhole mirror 75 by lens 41 and finally focused by relay lens 45 on a reference area RF of imaging sensor 60.

System Calibration.

As known, optical reflectometry allows measure certain sample's parameters that affect to its reflectance, for example stack layer thickness in CMP control or CD in photolithography control. In general, measurement process possible to divide into two stages:

• Measurement some reflected light characteristic, for example, light intensity spectrum, or interferogram, or phase difference spectrum, etc.

• Interpretation - finding the optimal parameters of measured sample mathematic model (theoretic characteristic), ensure the best fitting to measured characteristic.

In order to calculate theoretic characteristic, it's necessary to take into consideration optical property of measurement system, such as light source intensity spectrum, optical elements transmittance, sensor spectral response, etc. For this purpose system calibration process is required. For PSI systems calibration we claim two alternative approaches:

.

• System interferogram calibration.

The both approaches involve using calibration sample with known reflectance characteristics.

System Spectrum Calibration Methods.

In general, the PSI System Spectrum Calibration Methods and their using in measurement process have the following stages: a) Using calibration mirror, for example, silicon wafer, receive system spectrum as function of wave length and scanning position of active element:

S _Sys χλ,u) = ^Scλλ ^ (8.1), where

S _CM (λ, ύ) - light intensity spectrum, received on calibration mirror,

R _CM (λ) - known reflectance spectrum of calibration mirror.

The methods of receiving the light intensity spectrum on calibration mirror, are described bellow. b) Perform measurement cycle by scanning the active element position and receive interferogram on measured sample. c) Perform measured signal interpretation using mathematic model for interferogram or spectrum:

• Theoretic interferogram of light reflected from sample is calculated as:

I _x ^T (u,X) = jS _Syst (λ,u)x R _x ^τ (λ,X)dλ (8.2), where o

X - vector of measured parameters

i?J (λ, X) - theoretic reflectance spectrum.

Theoretic interferogram is fitted to measured interferogram during interpretation.

.

• Measured mterrerogram is converted to measured spectrum

The theoretic spectrum is calculated as:

Sl (λ,ϊ) = S^(λ,u) x R _x ^τ (λ,X) . (8.3)

Using Tunable Monochromatic Filter (TMF) for System Calibration.

PSI system could be calibrated using TMF with known spectral transmittance T _mF (λ) , that can be put in different places of the system. Fig. 22 shows example of adding TMF after light source 30. In that embodiment light is first focused by lens 39 on TMF, and collimated by lens 42 then reflected by a beam splitter 50 and focused on a sample (wafer) 100 by objective lens 40.

Returned from sample 100 (diffracted) light is than re-collimated by the objective 42, passing through the FF Filter 200 and finally focused on an imaging sensor 60 by a relay lens 41.

System spectrum calibration procedure could be performed as the follows:

• Put calibration mirror with known reflectance spectrum R _CM (λ) , on lens 40 focus plane (instead of sample 100).

• For different active element positions u _k receive light intensity image with different wave length λ _n (perform TMF scanning) I _CM (u _k , λ _n , x, y)

• Calculate common system spectrums for each imaging sensor pixel:

s _Sy λu _k ,λ,,, ^χ ,y) = ¹ CM («jtA > x, y)

^R CM OU ^{" T} TMF (.K ) In accordance with another broad aspect of the present invention it is possible to consider PSI systems as imaging Fourier spectrometer, based on interferential polarizational filter. The following calibration methods uses this internal spectrometer for system spectrum determination. The possible scheme for method implementation is shown on Fig. 18 A. The calibration procedure consists of two parts - active element phase delay calibration and system spectrum calibration using internal Fourier spectrometer.

According to another broad aspect of the present invention, in order to use internal Fourier spectrometer for system spectrum calibration, the active element phase delay calibration is

require . ne in erna in e eren ia po ariza iona i er rans ance, as runcuon or wave ieng n and scanning parameter value - u, can be expressed as:

T(X,u) = l ±cosφ(X,u) (8.4)

"+" for parallel polarizers orientation, "-" for crossed orientation. φ(λ, ύ) - phase delay, produced by active element.

For active element, implemented by Pockels cell or Soleil-Babinet (SB) compensator, the phase delay as function of wave length and scanning parameter can be expressed as: φ(X,u) = 2π - A(X) - U (8.5), where A(X) is function of wave length only, depended on active element optical properties. In order to find this function the following active element calibration procedure should be performed:

• Using reference spectrometer receive measured spectrum St?p (λ, u _k ) for different values of active element scanning parameter u _k .

• For each wave length X _n find A _n = A[X _n ) by fitting the measured spectrum to theoretical, calculated as:

S _S ^τ _P (λ _n ,u _k ) = S _ϋ (\ ± cos2πA _n u _k ) (8.6)

"+" for parallel polarizers orientation, "-" for crossed orientation.

According to stil another embodiment of the present invention, system spectral calibration could be performed using internal Fourier spectrometer. In that case after performing the active element phase delay calibration, it's possible to measure the PSI system spectrum by it's own Fourier spectrometer. The procedure is the following: • Put calibration sample, for example silicon wafer, with known reflectance spectrum R _Chi (X) to the focus plane of objective lens (see fig. 18.a).

scanning parameter, in other words, the matrix of interferograms I _CM (u _k ,x,y) .

• Using spectrum reconstruction algorithm, discribed in ... , receive spectrum image S _CM (λ, x, y) .

• Calculate system spectrum for each pixel of imaging sensor:

s _Sysl (λ x, y) -

Additionally, combined active element for system spectral calibration could be used. In order to receive enouthg high spectral resolution of Fourier spectrometer for system calibration, its active element should be able to produce relative high maximim OPD (more than 10 micron). Active elements with moveble parts, for example, Soleil-Babinet or Berek compensator, can produce such OPD, but from other side, are relative slow. If measurement task requires high throughput but doesn't require high scanning range at measurement stage, it's possible to use combined active element, comprised of two sequently arranged parts:

• high scanning range slow part for system calibration; • short scanning range fast part for measurement, implemented, for example by Pockels cell.

Fig. 23 exemplifies such a system.

At calibration stage, fast active element 20' is set to it's zero position and produces zero OPD. The system calibration is performed with scanning of the slow active element 20" as previously discribedi

At the measurement stage the fast active element 20 'is used for fast scanning of OPD.

System Interferogram Calibration Method.

The interferogram calibration method as well as spectral calibration methods, described above, uses interferential polarizational filter model, but doesn't use spectrum reconstruction algorithm. The method is based on extantion of the Convolution Theorem for interferential polarizational filter model, that takes into consideration the Time Delay (TD) dispersion in active element.

As known, the Convolution Theorem for COS Fourier transform looks as the following.

If f(ώ) = /, (ω) x / ₂ (ω) (multiplication), and F ₁ (t) and F ₂ (t) are COS Fourier transformations of /, (ω) and / ₂ (ω) correspondantly ( F ₁ (t) = F ₀ {/, (ω)} , F ₂ (t) = F _c {f ₂ (ω)} ) , therefore, COS Fourier transform of f(ω) can be received as convolution of F ₁ (0 and F ₂ (t) :

F(O = F _c {f(ω)} = F ₁ (0 * F ₂ (0 = -L ϊF _{ (t-τ) - F ₂ (τ)dτ (8.7)

If TD T , produced inside Fourier filter, is the same for all wave lengths (or angle frequencies ω ), for example, Michelson interferometer, the Convolution Theorem for centered interferograms looks as:

If I ₁ (τ) = JS ₁ (ω) cos ωdωτ and I ₂ (r) = JS ₂ (ω) cos ωdωτ , therefore,

1 -

J ₁₂ (T) = fa (O))S ₂ (ω) cos ωdωτ = —L (τ) * I ₂ (τ) (8.8)

Convolution Theorem Extension in Case of TD Dispersion. The active element of interferential polarizational filter produces TD τ = τ(ω,u) , depended on angle frequency ω and scanning parameter u (see part 9).

The centered interferogram, as function of scanning parameter, can be expressed as:

I(u) = js(ω) cos ωτ(ω,u)dω (8.9)

As shown in part 9.2.1, it's possible to consider the real Fourier spectrometer with time delay dispersion τ(ω,u) for regular spectrum S(ω) , as Fourier spectrometer without time delay dispersion for virtual spectrum S(ω) . Therefore, the centered interferogram can be expressed also via virtual spectrum:

/(T ₀ ) = jS(ω)cosωτ ₀ dω (8.10)

0

υ y u spciMmi

Let S ₁ (O)) and S ₂ (^y) are the regular spectrums, S ₁ (ω) and S ₂ (ω) are correspondent virtual spectrums. The regular and virtual spectrums differential congruence for the first spectrum looks as (see (9.16) equation): S _ι (ω)dω = S ₁ (ω)dω (8.11).

The (9.18. a) equation for S ₂ (ω) looks as:

S ₂ (ω) = S ₂ (ω)ω'(ω) (8.12).

Multiplication of (8.11) and (8.11) gives:

S ₁ (ω)S ₂ (ω)dω = S _λ (ω)S ₂ (ω)ω'(ω)dω (8.13). Let

S ₂ (ω) = S ₂ (ω)ω'(ω) (8.14).

Therefore,

S ₁ (ω)S ₂ (ω)dω = S ₁ (S)S ₂ (ω)dω (8.15).

Centered interferogram for real spectrums multiplication in case of time delay dispersion looks as:

J ₁₂ (τ ₀ ) = JS ₁ (ω)S ₂ (ω) cos ωτ(ω, τ ₀ )dω (8.16).

Therefore, for virtual spectrums it looks as regular cos Fourier transform of two function multiplication:

J ₁₂ (r ₀ ) = ψ _x (ω)S ₂ (ω) cos ωτ _o dω (8.17).

Applying the regular Convolution theorem for cos Fourier transform (see (8.8) equation) gives:

_ ^∞ ~ 1 - _

40o) = Fi ( ^® )$2 ( ^® ) ^{cos ®τ} o ^dS > = - ^J ι(h) * I ₂ (To) ( ⁸ - ¹⁸ )» o ^π where

J ₁ (T ₀ ) = JiS ₁ (ω) cos ωdωτ ₀ (8.19. a), and

^ ₂ ( ^r _o ) = J^ ₂ (S) cos ωdωτ ₀ (8.19.b) o are centered interferograms of correspondent virtual spectrams. For real spectrums the (8.19. a), and (8.19.b) looks as:

J ₁ (T ₀ ) = JiS ₁ (O)COS Or(OjTo)Cf(B (8.20. a), and

I ₂ (T ₀ ) = \s ₂ (ω) cos COT (ω,τ _Q )dω (8.20.b).

Measurement Procedure generally includes three main steps: Calibration.

• Put calibration sample, for example silicon wafer, with known reflectance spectrum R _CM (o)to the focus plane of objective lens (see Fig. 18. a).

• Receive set of light intensity images for certain values of active element scanning parameter, in other words, the matrix of interferograms I _CM (u _k ,x,y) .

• Calculate and save centered interferograms matrix I _cω (u _k , x, y)

Measurement.

• Put measurement sample to the focus plane of objective lens (see fig. 18. a).

• Receive set of light intensity images for certain values of active element scanning parameter, in other words, the matrix of experimental interferograms Interpretation.

oi caυπ A aiiu y nn e measure parame er vector , a ensures e best fitting or e experimental centered interferogram Tf^ (τ _Ok ) to theoretical interferogram /J'""" ^" (τ _ok , X) .

The theoretical interferogram for each X can be calculated by the following sequence:

• Receive theoretical reflectance spectrum R™" ^01' (ω,X) from mathematic model or library.

• Calculate relative theoretical reflectance spectrum:

R? ^e ° ^r {ω,X) = ^{Rχ iω} ' ^X) (8.21).

^R CM O)

• Calculate theoretical centered interferogram for relative reflectance spectrum, that considers time delay dispersion of real Fourier filter:

l _κ (γ ₀ , X) = ωτ(ω, T ₀ )dω (8.22).

• Calculate the resulting theoretical centered interferogram, as convolution of calibration interferogram (see 8.2.3.1) and (8.22):

I^ ^or (τ ₀ ,X) = -I _cu (τ ₀ ) * I _R (T ₀ , X) (8.23).

TC '

According to another embodiment of the present invention, PSI Calibraiton could be perfomed without Fourier Filter Modeling and Spectrum Reconstruction

This calibration method doesn't require PSI Fourier filter modeling, and is discribed in attched document

Spectrum Reconstruction Algorithm. The direct signal of all Fourier spectrometers is interferogram, which is the function of integral light intensity vs. scanning parameter:

I(u) = \S(ω) x [1 ± cos ωτ(u, ω)]dω (9.1), where o

S(ω) - measured light spectrum vs. angular frequency,

_? _ scanning parameter and, in general case, also of angular frequency.

Notice, that sign "+" or "-" in equation (9.1) depends on Fourier spectrometer type and configuration. Light intensity spectrum can be exstructed from interferogram using spectrum reconstruction algorithm.

In case of negligibly small time delay dispersion, for example in the Michelson interferometer, it's possible to use the standard spectrum reconstruction algorithm, based on expansion into Fourier series. In case of PSI, the standard spectrum reconstruction algorithm brings the notable error because of time delay dispersion. The claimed spectrum reconstruction algorithm takes into consideration the delay dispersion and allows serious to decrease this error.

The Standard Spectrum Reconstruction Algorithm Description.

In case of negligibly small time delay dispersion, the interferogram (see equation (9.1)) can be expressed as function of time delay:

/(T) = jS(ω) x [1 + cos ωτ]dω (9.2)

0

Therefore, the centered interferogram is the cos Fourier transform of spectrum:

J(τ) = Js(O) cos ωτdω (9.3) o

The centered interferogram can be find as: 7(r) = ±[ J(T) - Z ₀ ] (9.4),

where I ₀ = $S(ω)dω (9.5).

For real light intensity spectrums it's possible to select the max frequency ω _max that S(ω) « 0 for ω > ω _max . So the (9.3) equation looks as:

J(O = J5(α?) cos ωτdω (9.6)

As known, the expansion of symmetric function S(ω) on limited interval into Fourier series looks as:

where τ _n - n (Nyquist criterion) (9.8).

^max

Therefore, spectrum can be reconstructed from centered interferogram points, satisfied the Nyquist criterion (see equation (9.8)):

$(ω) = ^- + ∑-?—I(τ _n )cosτ _n ω (9.10).

Time Delay Dispersion Consideration in Spectrum Reconstruction Algorithm.

For many types of Fourier spectrometer the time delay has the same mathematic expression - multiplication of scanning parameter u and some angular frequency function k(ώ) , determined by active element optical properties. τ{u _i ώ) = k{ώ) 'x u (9.11).

It's relevant for PSI system, contained active element, implemented, for example, by Pockels cell or Soleil-Babinet compensator.

In genaral, the claimed algorithm is appropriated only for Fourier spectrometers with such type of time delay function.

Spectrum Transformation.

The (9.11) equation can be exspressed as:

24

SUBSTITUTE SHEET (lull; It)

φ,ω) = - -χ τ _o (u) (9.12),

where T ₀ (u) = k(ω _mxi ) x u - is time delay, corresponded to max angular frequency. Let's define the relative dispersion function as:

/<»> = #*r (9-13).

Therefore, interferogram, as function of τ ₀ , can be expressed as:

/(T ₀ ) = jS(ω) x [1 ± cos(/(ω) - ^y - T ₀ )]dω (9.14). o

Let's define virtual angular frequency as: ω = ω - f(ω) (9.15),

and and virtual light intensity spectrum S(S) , satisfied the following differential condition:

S(ω)dω = S(ω)dω (9.16).

It means that light intensity, localized inside interval ω í ω + dω of virtual spectrum S(ω) , equals to light intensity, localized inside interval ω í ω + dω of regular spectrum S{ώ) .

Fig, 24 further illustrates the above described principal

The integral light energies (interferogram), expressed via regular and virtual spectrums, are equaled:

I(τ ₀ ) = jS(ω) x [1 ± cos(f(ω) - ω - τ ₀ )]dω = JS(S) x [1 ± cos ωτ ₀ ]dω (9.17).

0 0

The equation of differentials congruence (9.16), allows to connect real and virtual spectrums as:

S(ω) = S(ω(ω))^^- (9.18.a) dω S(ω) = S(ω(ω)) x \f(ω) + cof'(ω)] (9.18.b).

_{0 ec rome er w} p n r _{0 O} r regular spectrum S(ω) , as Fourier spectrometer without time delay dispersion for virtual spectrum S(ω) .

The spectrum reconstruction algorithm that takes into consideration time delay dispersion appears as follows:

• Calculate scanning parameter volues u _n , that ensure the matching to Nyquist criterion of time delay for maximal angular frequency ω _msx :

φ,,,ω _m!Si ) = τ _n0 = n— (9.19).

• Measure the interferogram points for calculated volues of scanning parameter:

I _n = I(Uj = I(T ₁₁₀ ) (9.20).

• Using the (9.15) equation, calculate virtual maximal angular frequency:

• Using reconstructed spectrum expression of the standard spectrum reconstruction algorithm (see (9.10) equation), calculate virtual spectrum:

S(ω) = -I ₀ )cosωτ _n0 (9.22).

• Using (9.18.b) equation, transform the virtual spectrum & (ω) to real spectrum S(ω) .

While the subject invention has been described with reference to number of preferred embodiments, various changes and modifications could be made therein, by one skilled in the art, without varying from the scope and spirit of the subject invention as defined by the appended claims.